My preference would be to first decompose the notes field into its parts (all of them) and to store it into some structure (like a hash) and only then extract the information you are looking for.

Here, a split would do exactly that using captures in the regex to keep the field names.

use strict;
use warnings;
use Data::Dumper;

while(<DATA>){
    my( $notesField )= /"(.*)"/;
    my @parts = split /\s?(First Name|Last Name|Address|City|State|ZIP
+ Code|E-mail): /, $notesField;
    shift @parts;
    my %parts = @parts;
    print Dumper \%parts;
}
__DATA__
,,,,,,,,,,,,,,,,,,,,,,,,,"First Name: Dobbin Last Name: David L. Addre
+ss: david@adamsonanddobbin.com City: PO Box 1326407 Pido Road State: 
+Peterborough ZIP Code: ON Country: K9J 7H5 First Name: Dobbin Last Na
+me: David L. E-mail: david@adamsonanddobbin.com Address: PO Box 13264
+07 Pido Road City: Peterborough State: ON ZIP Code: K9J 7H5",,,,,,Hom
+e,743 7790,Other,742 4524,Work,745 5751,,,,,,,,,,,Adamson And Dobbin 
+Ltd. Mechanical Contractors,,General Manager,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,,,,,,"First Name: Chapleau Last Name: Kathy, Ken A
+ddress: 666 FrankFirst Name: Chapleau Last Name: Kathy, Ken City: 666
+ Frank",,,,,,Home,876-9863,,,,,,,,,,,,,,,Admiralty Hall,,Accountant,,
+,,,,,,,,,
[download]

The result would be a hash like this:

$VAR1 = {
          'First Name' => 'Dobbin',
          'ZIP Code' => 'K9J 7H5',
          'Address' => 'PO Box 1326407 Pido Road',
          'Last Name' => 'David L.',
          'City' => 'Peterborough',
          'E-mail' => 'david@adamsonanddobbin.com',
          'State' => 'ON'
        };
$VAR1 = {
          'First Name' => 'Chapleau',
          'Address' => '666 Frank',
          'Last Name' => 'Kathy, Ken',
          'City' => '666 Frank'
        };
[download]

Please note that my extraction of the notes field is only done using a regex for convenience. Your approach using Text::CSV is clearly the right way to do it.

When using a hash you would lose any duplicates. So if there are two first names, only one of them would survive.